Degrees of Freedom Help: Comparing Fits to Data with Unknown X Values

[cahill8]

I'm trying to compare fits to data in a certain way. I start with the initial data and fit and compute the $\chi$^2 value for example:

x: 1 2 3 4
y: 2.1 3.9 6.0 10.1

I get the linear fit, $\chi$^2 value and degrees of freedom=3.

Now with what I'm working with, I don't actually know the values of x. The above values of x are just assumed (but the y values are known). I'm looking to reduce the $\chi$^2 value so I let the fourth x value (4) equal 5 instead and then make a new linear fit with a new $\chi$^2 value. It's obvious that the new $\chi$^2 value will be smaller than the original but do I lose a degree of freedom by choosing the last x value to be 5 instead of assuming its 4?

I want to be able to compare the $\chi$^2/(degrees of freedom) value for the original case (final x=4) and the adjusted case (final x=5).

I'm confused for either case. E.g. If the degrees of freedom remain the same, I can keep adding more and more points to get a better fit without any penalty.
On the other hand, if the degrees of freedom are reduced, if I were to add more points, would the degrees of freedom eventually hit 0?

 

[mmwave]

it is important to carefully consider and analyze all aspects of your data and methods. In this case, your approach to comparing fits to data is a valid one, but there are a few points to consider.

First, let's clarify the concept of degrees of freedom (dof). In general, the degrees of freedom for a fit to data is equal to the number of data points minus the number of parameters in the fit. In your example, you have 4 data points and 2 parameters (slope and intercept) in your linear fit, so the dof is 2. This means that you have 2 independent pieces of information that can be varied in your fit.

Now, let's address your main question about changing the last value of x from 4 to 5. This does not necessarily change the degrees of freedom. In fact, in your example, it would not change the dof because you are still fitting 4 data points with 2 parameters. However, it is important to note that changing the value of x can affect the quality of your fit and the resulting \chi^2 value. In general, a smaller \chi^2 value indicates a better fit to the data.

Next, let's address your concern about adding more points and potentially reducing the degrees of freedom to 0. In theory, this could happen if you continue to add more and more points to your fit. However, in practice, it is unlikely to occur because there will always be some degree of uncertainty or error in your data, which will prevent the dof from reaching 0. Additionally, it is important to keep in mind that adding more points does not necessarily lead to a better fit. It is important to carefully consider the quality and relevance of the additional data points before including them in your fit.

In summary, your approach of comparing \chi^2/(degrees of freedom) values for different fits is a valid way to assess the quality of your fits. Just be sure to carefully consider the impact of changing the values of your parameters and adding more data points, and always keep in mind the concept of degrees of freedom and its implications for your analysis.